EXCLUSIVE INTERVIEW WITH RICHARD TAYLOR,
including
Math-123-level overview of his and Wiles' proof of FLT.

When a mistake was found in Andrew Wiles' original proof, he called in
Cambridge algebraist Richard
Taylor. They worked together at Princeton for eight or nine months and
emerged with a solution to the world's most famous open problem. Taylor is
now
at Harvard and will be teaching Math 121 in the spring.

RICHARD TAYLOR,
interviewed by Scott
Sheffield

HOW LONG HAVE YOU BEEN AT HARVARD? Just over two
weeks.

WHERE ARE YOU FROM? I spent most of my career in Cambridge, but
last year
I moved to Oxford for one year before coming here. Before that I did a one
year
post-doc in Paris, a PhD in Princeton and was an undergraduate in
Cambridge.

WHAT ARE YOUR MAJOR RESEARCH INTERESTS AND ACHIEVEMENTS? The great
problem that motivates me is to understand the absolute Galois group of
the
rational numbers, that is, the group of all automorphisms of the field of
algebraic numbers (complex numbers which are the roots of nonzero
polynomials
with rational coefficients). If you like you can talk about all Galois
groups of
finite extensions of the rational numbers, but this is a convenient way to
put
them all together. It doesn't make a lot of difference, but it is
technically
neater to put them all together. The question that has motivated almost
everything I have done is, "What's the structure of that group?" One of
the
great achievements of mathematicians of the first half of this century is
called
class field theory, and one way of seeing it is as a description of all
abelian
quotients of the absolute Galois group of Q, or if you like, the
classification
of the abelian extensions of the field of the rational numbers. That's
only a
very small part of this group. The group is extremely complicated, and
just
describing the abelian part doesn't solve the problem. For instance John
Thompson proved that the monster group is a quotient group of this group
in
infinitely many ways.

There is some sort of program to understand the rest of this group,
often
referred to as the Langlands Program. There's a huge mass of conjectures,
of
which we are only beginning to scratch the surface, which tell us what the
structure is. The answer is to my mind extremely surprising; it invokes
extremely different objects. You start out with this algebraic structure
and end
up using what are called modular forms, which relate to complex analysis.

There seems to be an answer to this question: what's the structure? And
the
answer is something completely unexpected in terms of these analytic
objects,
and I think that's what attracts me to the subject. When there is a great
connection between two different areas of mathematics, it always seems to
me
indicative that something interesting is going on.

The other thing we can see--another indication that it's a powerful
theory--is that one can answer questions one might have asked anyway,
before one
built up the theory. Maybe, the first example was a result proved by Barry
Mazur; he provided a description of the possible torsion subgroups of
elliptic
curves defined over the rational numbers. It was a problem that had been
knocking around for some time, and it's relatively easy to state. Using
these
sorts of ideas, Barry was able to settle it.

Other examples are the proof the main conjecture of Iwasawa theory by
Barry
Mazur and Andrew Wiles, and the work of Dick Gross and Don Zagier on
rational
points on elliptic curves. And I guess finally, there's Fermat's last
theorem,
which Andrew Wiles solved using these ideas again. So in fact, the story
of
Fermat's last theorem is that this German mathematician Frey realized that
if
you knew enough of this correspondence between modular forms and Galois
groups,
there is an extraordinarily quick proof of Fermat's last theorem. And at
the
time he realized this, not enough was known about this correspondence.
What
Andrew Wiles did and Andrew and I completed was prove enough about this
correspondence for Frey's argument to go through. The thing that amuses me
is
that it seems that history could easily have been reversed. All these
things
could have been proved about the relationship between modular forms and
Galois
groups, and then Frey could have come along and given nearly a two-line
proof of
Fermat's last theorem.

Those four [torsion points, Iwasawa theory, Gross and Zagier, Fermat]
are
probably the obvious big applications of these sorts of ideas. It seems to
me
the applications have been extraordinarily successful--at least four
things that
would have been recognized as important problems irrespective of this
theory,
problems that people had thought about before modular forms.

Somehow applications of this theory have been going for some
twenty-five
years. Barry's result was in the early seventies. (I think.)

HOW OLD ARE MODULAR FORMS? Certainly about thirty years. Sort of in
this
period the ideas have been becoming more and more fixed. The first
indications
maybe go back to maybe the late fifties. But the ideas didn't really start
becoming definite until maybe 1970. These dates are very rough.

WHAT ARE MODULAR FORMS? Modular forms are holomorphic functions
defined
on the upper half complex plane--only the part with positive imaginary
part. The
group SL_2(Z) acts on the upper complex plane by Mobius transformation; by
composition, the group also acts on the set of holomorphic functions of
the
upper complex plane. Modular forms are functions which transform in a
simple way
under the action of that group.

WHAT WAS YOUR PERSONAL ROLE IN THE DEVELOPMENT OF MODULAR FORMS?
I've
done various things, but they're all rather difficult to explain on this
sort of
level. Maybe the simplest thing to talk about is the following: There
should be
some sort of correspondence between certain of these modular forms and
two-dimensional representations of the absolute Galois group of Q. In one
direction, things have been known for twenty-five years or something. If
one
starts with a suitable modular form, the way to construct a representation
of
the Galois group has been known for twenty-five years. Now the big problem
has
been to start with a representation of the Galois group and try to produce
a
modular form. In fact, there's one result that's rather old, due to
Langlands
and Tunnell. Until rather recently, that has been the only isolated
result.
Recently, Andrew Wiles did much better. I guess I was involved with this
in the
end. It is probably well known that there was a mistake the first time he
tried
to do something like this. He spent a few months trying to fix the mistake
himself. Then he rang me up one day and to my great surprise asked me to
come
help him work on the problem. We worked together for eight or nine months
and
eventually found a way to get the arguments to work.

And I guess before that, my main interests have been in certain
generalizations of these questions. For instance, if instead of the
rational
numbers one took the Gaussian numbers, one can ask the same sorts of
questions.
It's slightly less obvious there what one should mean by a modular form.
They
turn out to be functions not on the upper complex plane but on hyperbolic
three-space. So I spent a lot of time trying to copy as much of what was
known
for the rational numbers to other fields like Gaussian numbers and Q
adjoined
the root of a negative number, say Q(root(-d)).

So a lot of other people have thought about totally real fields. A
totally
real field is a finite extension of the rational numbers such that
whenever you
embed it in the complex numbers, it actually lies in the real numbers.
Q(root(2)) is an example. Q(cube-root(2)) is not an example, it can be
embedded
entirely in R, but it doesn't have to be. It turns out that totally real
fields
seem easiest for this theory. I thought about these for a bit. Then I
turned to
things like the Gaussian integers, Q(i), which are the simplest examples
of
non-totally real fields. This is probably what I was best known for in our
little circle for before the work on Fermat.

Classical modular forms are holomorphic. There is no notion of
functions on
hyperbolic three-space being holomorphic. It's not a complex space--it's
got
three real dimensions. It's this lack of being able to talk about things
that
are holomorphic that make this case and anything that isn't totally real
harder.

WHAT ARE YOUR MAJOR RESEARCH GOALS FOR THE FUTURE? Certainly at the
moment I'm thinking about the same sort of questions. This solution of the
Fermat conjecture got so much publicity, but in a sense it's only a small
way
towards the goal of working out this correspondence between
representations of
Galois groups and modular forms and their generalizations. There is far
more
left to be done than has been done. There have been some big steps
forward, but
compared to what's left to do, there is still an awful lot left to do.
We're
only scratching the surface. To a large extent, we feel confident that we
know
what's true, but we're very far from proving most of it. It's very
tantalizing,
this big, beautiful picture that we can't get our hands on.

ARE YOU COLLABORATING WITH OTHER FACULTY AT HARVARD? At the moment
I'm
working by myself, but I've only just arrived here. It's certainly a great
place
to do this sort of this thing. Barry Mazur, Dick Gross, Noam Elkies--you
couldn't ask for a better group of colleagues in our subject.

WHAT COURSES DO YOU HOPE TO TEACH IN THE SPRING? In the spring I'm
teaching Math 121. I've yet really to discover what's in the course or
anything.
I am looking forward to teaching math majors in future years, and I'm sure
I
will. I'm sure I'll teach a variety of things, algebra, algebraic
geometry,
number theory. I'm sure I'll be teaching graduate courses.

DO YOU PLAN ON ADVISING ANY UNDERGRADUATE STUDENTS FOR THESES?
Chris
Degni came to see me. He's doing something on some conjectures of Serre in
this
area. Senior theses are something that doesn't exist in England. It's a
concept
that's new to me, so I'll have to learn what's expected. Four graduate
students
have moved with me from England here, so I have four graduate students.
Three of
them that are relatively early in their graduate career and will get
Harvard
degrees, the fourth is in his final year and will get an Oxford degree.

WHY DID YOU CHOOSE TO COME TO HARVARD? I guess I got the formal
offer in
the spring from the dean, but we'd obviously talked about it with the
faculty
here for some time before that. One strong personal reason is that my
wife's
American and would like to be in America. Also it's a great department.
Like I
say, it's difficult to imagine a better collection of colleagues in my
subject
than there is here. By all accounts, the students here are very bright. I
don't
really have personal experience, but I'm sure it's true.

WHAT DO YOU LIKE/DISLIKE ABOUT HARVARD LIFE? I actually visited for
six
months a couple of years ago, and one thing I like is the sun. Somehow in
Britain for half the year, it's extraordinarily dark. That's partly
because it's
further north and partly because there is more cloud cover. I've heard
people
complain that in the winter it's cold here, but at least you see the sun.
And I
like the energy; people are very energetic and enthusiastic here.
Something I
noticed is that in Britain it's cool to pretend you never do any work.
Students
there obviously do work because they learn the same stuff as anybody else,
but
they like to pretend they do nothing. Whereas here, people in Princeton
would
come to me and tell me they had spent the last twenty-four hours in the
library.
Here, they seem to pretend they work harder than they do. I suspect that
people
work the same in both places; it's just the gloss they put on it.

This department is an extremely friendly department. People just seem
to talk
to each other more than they do in many places.

HOW DOES THE AMERICAN SYSTEM OF EDUCATION COMPARE TO THE BRITISH
SYSTEM?
Undergraduates in England usually study one subject. Most mathematics
students in Cambridge are only studying mathematics; they spend 100% of
their
time studying that. This makes teaching there a different experience from
teaching people who are studying mathematics as part of a broader
education. I
have the impression that most teaching here is done in middle sized
classes. In
Cambridge there is a combination of very large classes--100 people or so--
or
very small classes where one or two students meet with one professor. The
continuous assessment is also different. In England, the only assessment
is at
the end of the year. Through the year you get no grades at all, and
everything
depends on how your perform over two days during the large exam at the
end. I
don't yet have enough experience with the American system to know which I
prefer, but there are these differences of style.

WHEN DID YOU BECOME INTERESTED IN MATH? Very early, I suspect. My
father
is a theoretical physicist. There was always a culture of mathematical
science
in the family. I don't remember exactly, but certainly as a teenager I was
interested in mathematics. I just enjoyed reading recreational books on
mathematics and trying to do math problems and finding out about more
advanced
mathematics. There wasn't any one thing that struck me as particularly
interesting.

WHEN DID YOU FIRST DISCOVER YOU HAD TALENT IN MATHEMATICS? Well, I
guess
already in high school it was clear that I was better than most of the
other
kids in mathematics. But as you go on, you're always mixing with people
who are
more talented in mathematics. It is never clear if you have a real talent
or
just appear talented in the group you are currently mixing with. I really
enjoy
mathematics. I think great interest in mathematics and determination to
persevere accounts for more than people often give credit for. If you are
very
keen on working on mathematical problems, you usually get good at it, and
I
think this can make up for a fair amount of mathematical talent. I have
certainly know people who are far brighter mathematicians than I am, but
if they
have thought about a problem for two days and can't solve it, they get
bored
with it and want to move on. But that is not a recipe for good research;
you
have to just keep going on and on.